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A novel hybrid whale–Nelder–Mead algorithm for optimization of design and manufacturing problems

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Abstract

This paper introduces a new hybrid optimization algorithm (HWOANM) based on the Nelder–Mead local search algorithm (NM) and whale optimization algorithm (WOA). The aim of hybridization is to accelerate global convergence speed of the whale algorithm for solving manufacturing optimization problems. The main objective of our study on hybridization is to accelerate the global convergence rate of the whale algorithm to solve production optimization problems. This paper is the first research study of both the whale algorithm and HWOANM for the optimization of processing parameters in manufacturing processes. The HWOANM is evaluated using the well-known benchmark problems such as cantilever beam problem, welded beam problem, and three-bar truss problem. Finally, a grinding manufacturing optimization problem is solved to investigate the performance of the HWOANM. The results of the HWOANM for both the design and manufacturing problems solved in this paper are compared with other optimization algorithms presented in the literature such as the ant colony algorithm, genetic algorithm, scatter search algorithm, differential evolution algorithm, particle swarm optimization algorithm, simulated annealing algorithm, artificial bee colony algorithm, improved differential evolution algorithm, harmony search algorithm, hybrid particle swarm algorithm, teaching-learning–based optimization algorithm, cuckoo search algorithm, grasshopper optimization algorithm, salp swarm optimization algorithm, mine blast algorithm, gravitational search algorithm, ant lion optimizer, multi-verse optimizer, whale optimization algorithm, and the Harris hawks optimization algorithm. The results show that the HWOANM provides better exploration and exploitation properties, and can be considered as a promising new algorithm for optimizing both design and manufacturing optimization problems.

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  1. Department of Automotive Engineering, Faculty of Engineering, Bursa Uludag University, Görükle, 16059, Bursa, Turkey

    Ali Riza Yildiz

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Appendix 1

Appendix 1

1.1. Three-bar truss problem

The problem can be mathematically formulated as follows:

Minimize: \( f(x)=\left(2\sqrt{2{A}_1}+{A}_2\right)\times l \)

Subject to:

$$ {g}_1(x)=\frac{\sqrt{2}{A}_1+{A}_2}{\sqrt{2}{A}_1^2+2{A}_1{A}_2}p-\sigma \le 0,\kern6em {g}_2(x)=\frac{A_2}{\sqrt{2}{A}_1^2+2{A}_1{A}_2}P-\sigma \le 0 $$
$$ \kern0.5em {g}_3(x)=\frac{1}{\sqrt{2}{A}_2+{A}_1}P-\sigma \le 0 $$

where 0 ≤ A1 ≤ 1, 0 ≤ A2 ≤ 1, l = 100 cm, P = 2kN, σ = 2kN/cm2

1.2. Cantilever problem

The problem can be mathematically formulated as follows:

Minimize :

$$ f(X)=0.0624\left({x}_1+{x}_2+{x}_3+{x}_4+{x}_5\right) $$

Subject to:

$$ \mathrm{g}(x)=\frac{61}{x_1^3}+\frac{37}{x_2^3}+\frac{19}{x_3^3}+\frac{7}{x_4^3}+\frac{1}{x_5^3}-1\le 0 $$

The design variables are the heights (or widths) of the different beam elements, and the thickness is held fixed (here t = 2/3). The bound constraints are set as 0.01 ≤ xj ≤ 100. This problem can be expressed analytically as follows:

1.3. Welded beam design problem

The problem can be mathematically formulated as follows:

Minimize: f(x) = 1.10471h2l + 0.04811tb(14.0 + l)

Subject to:

$$ {g}_1(x)=\tau (x)-{\tau}_{\mathrm{max}}\le 0,\kern5.25em {g}_2(x)=\sigma (x)-{\sigma}_{\mathrm{max}}\le 0 $$
$$ {g}_3(x)=h-b\le 0,\kern8.25em {g}_4(x)=0.1047{h}^2+0.04811 tb\left(14+l\right)-5\le 0 $$
$$ {g}_5(x)=0.125-h\le 0,\kern6.25em {g}_6(x)=\delta (x)-{\delta}_{\mathrm{max}}\le 0 $$
$$ {g}_7(x)=P-{P}_c(x)\le 0 $$

where \( \tau =\sqrt{{\left({\tau}^{\prime}\right)}^2+{\left({\tau}^{\prime \prime}\right)}^2+2{\tau}^{\prime }{\tau}^{\prime \prime}\frac{l}{2R}},\kern0.75em {\tau}^{\prime }=\frac{P}{\sqrt{2} hl},\kern0.75em {\tau}^{\prime \prime }=\frac{MR}{J},M=P\left(L+\frac{1}{2}\right),R=\sqrt{\frac{l^2}{4}+\frac{{\left(h+t\right)}^2}{4}} \)

$$ J=2\left\{\sqrt{2} hl\left[\frac{l^2}{12}+\frac{{\left(h+t\right)}^2}{4}\right]\right\},\sigma =\frac{6 PL}{bt^2},{P}_c=\frac{4.013E\sqrt{t^2{b}^6/36}}{bt^2}\left(1-\frac{t}{2L}\sqrt{\frac{E}{4G}}\right) $$
$$ P=6000 lp,l=14 in,E=30\times {10}^6 psi,G=12\times {10}^6 psi,{\tau}_{\mathrm{max}}=13600\ psi,{\delta}_{\mathrm{max}}=0.25 in,0.1\le h,b\le 2 $$

and 1 ≤ l, t ≤ 10.

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Yildiz, A.R. A novel hybrid whale–Nelder–Mead algorithm for optimization of design and manufacturing problems. Int J Adv Manuf Technol 105, 5091–5104 (2019). https://doi.org/10.1007/s00170-019-04532-1

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